Objectives Graphs of Sine and Cosine Graphs of Transformations of Sine and Cosine Using Graphing Devices to Graph Trigonometric Functions

Graphs of Sine and Cosine

Graphs of Sine and Cosine To sketch the graphs on the interval 0  t  2, we could try to make a table of values and use those points to draw the graph. To draw the graphs more accurately, we find a few other values of sin t and cos t in Table 2. We could find still other values with the aid of a calculator. Table 2

Graphs of Sine and Cosine Now we use this information to graph the functions sin t and cos t for t between 0 and 2 in Figures 2 and 3. Graph of sin t

Graphs of Sine and Cosine Graph of cos t These are the graphs of one period. Using the fact that these functions are periodic with period 2, we get their complete graphs by continuing the same pattern to the left and to the right in every successive interval of length 2.

Graphs of Sine and Cosine The graph of the sine function is symmetric with respect to the origin. This is as expected, since sine is an odd function. Since the cosine function is an even function, its graph is symmetric with respect to the y-axis.

Graphs of Transformations of Sine and Cosine

Example 1 – Cosine Curves Sketch the graph of each function. (a) f (x) = 2 + cos x (b) g (x) = –cos x Solution: (a) The graph of y = 2 + cos x is the same as the graph of y = cos x, but shifted up 2 units (see Figure 4(a)).

Example 1 – Solution cont’d (b) The graph of y = –cos x in Figure 4(b) is the reflection of the graph of y = cos x in the x-axis.

Graphs of Transformations of Sine and Cosine Let’s graph y = 2 sin x. We start with the graph of y = sin x and multiply the y-coordinate of each point by 2. This has the effect of stretching the graph vertically by a factor of 2. To graph y = sin x, we start with the graph of y = sin x and multiply the y-coordinate of each point by . This has the effect of shrinking the graph vertically by a factor of (see Figure 5).

Graphs of Transformations of Sine and Cosine In general, for the functions y = a sin x and y = a cos x the number | a | is called the amplitude and is the largest value these functions attain. Graphs of y = a sin x for several values of a are shown in Figure 6.

Example 2 – Stretching a Cosine Curve Find the amplitude of y = –3 cos x, and sketch its graph. Solution: The amplitude is | –3 | = 3, so the largest value the graph attains is 3 and the smallest value is –3. To sketch the graph, we begin with the graph of y = cos x, stretch the graph vertically by a factor of 3, and reflect in the x-axis, arriving at the graph in Figure 7.

Graphs of Transformations of Sine and Cosine Since the sine and cosine functions have period 2, the functions y = a sin kx and y = a cos kx (k > 0) complete one period as kx varies from 0 to 2, that is, for 0  kx  2 or for 0  x  2 /k. So these functions complete one period as x varies between 0 and 2/k and thus have period 2 /k. The graphs of these functions are called sine curves and cosine curves, respectively. (Collectively, sine and cosine curves are often referred to as sinusoidal curves.)

Graphs of Transformations of Sine and Cosine To see how the value of k affects the graph of y = sin kx, let’s graph the sine curve y = sin 2x.

Graphs of Transformations of Sine and Cosine Since the period is 2 /2 =, the graph completes one period in the interval 0  x   (see Figure 8(a)). For the sine curve y = sin x, the period is 2  = 4, so the graph completes one period in the interval 0  x  4 (see Figure 8(b)). Figure 8(a) Figure 8(b)

Graphs of Transformations of Sine and Cosine We see that the effect is to shrink the graph horizontally if k > 1 or to stretch the graph horizontally if k < 1. For comparison, in Figure 9 we show the graphs of one period of the sine curve y = a sin kx for several values of k. Figure 9

Example 3 – Amplitude and Period Find the amplitude and period of each function, and sketch its graph. (a) y = 4 cos 3x (b) y = –2 sin x Solution: (a) We get the amplitude and period from the form of the function as follows:

Example 3 – Solution The amplitude is 4 and the period is 2 /3. The graph is shown in Figure 10. cont’d Figure 10

Example 3 – Solution b) For y = –2 sin x cont’d amplitude = | a | = | 2 | = 2 period = = 4 The graph is shown in Figure 11. cont’d Figure 11

Graphs of Transformations of Sine and Cosine The graphs of functions of the form y = a sin k(x – b) and y = a cos k(x – b) are simply sine and cosine curves shifted horizontally by an amount | b |. They are shifted to the right if b > 0 or to the left if b < 0. The number b is the phase shift.

Graphs of Transformations of Sine and Cosine We summarize the properties of these functions in the following box.

Example 4 – A Shifted Sine Curve Find the amplitude, period, and phase shift of y = 3 sin 2 , and graph one complete period. We get the amplitude, period, and phase shift from the form of the function as follows:

Example 4 – Solution Since the phase shift is  /4 and the period is , one complete period occurs on the interval As an aid in sketching the graph, we divide this interval into four equal parts, then graph a sine curve with amplitude 3 as in Figure 13. cont’d Figure 13

Example 5 – A Shifted Cosine Curve Find the amplitude, period, and phase shift of and graph one complete period. Solution: We first write this function in the form y = a cos k(x – b). To do this, we factor 2 from the expression 2x + to get

Example 5 – Solution Thus we have amplitude = | a | = period = =  phase shift = b = From this information it follows that one period of this cosine curve begins at – /3 and ends at (– /3) +  = 2 /3. cont’d Shift to the left

Example 5 – Solution To sketch the graph over the interval [– /3, 2 /3], we divide this interval into four equal parts and graph a cosine curve with amplitude as shown in Figure 14. cont’d Figure 14